A Class-D audio amplifier is a switching amplifier or PWM amplifier. Class-D amplifier usually can provide high power efficiency over 90%, comparison to the 50% provided by conventional linear amplifier. To obtain a high-SNR Class-D amplifier, a feedback loop is often included. FIG. 1 shows a schematic view of a conventional Class-D amplifier. As shown in FIG. 1, Class-D amplifier is embodied by a PWM generator 102 and a noise shaping sigma-delta modulator 101, wherein the PWM generator 102 outputs complementary signals to a power driver 103 and through a filter 104 to drive a load. The drawback of the above embodiment is that sigma-delta modulation suffers stability problem and the modulator output signal gain is less than 1.
FIG. 2 and FIG. 3 show schematic views of a conventional PWM generator and corresponding waveform of the conventional PWM generator respectively. As shown in FIG. 2, the input of modulating digital audio signal S(θ)=B sin(θ), where 0≦B≦1, is modulated by a differential PWM generator. The PWM sample-rate is defined as ωc=Mωs, where M is an integer greater than 2. FIG. 3 shows a schematic view of the waveform of digital audio input and output signal Vo of the digital differential PWM, generator, where Vo=DP−DN, and output signal Vo can be expressed as Fourier series:V0(θ)=Σn=1∞[An cos(nθ)+Bn sin(nθ)]  (1)Where
                              A          n                =                  2          ⁢                      V            CC                    ⁢                                    ∑                              k                =                0                                            M                -                1                                      ⁢                                                            Sign                  ⁡                                      (                                          s                      ⁡                                              (                                                                              2                            ⁢                            π                            ⁢                                                                                                                  ⁢                            k                                                    M                                                )                                                              )                                                                    n                  ⁢                                                                          ⁢                  π                                            [                                                cos                  ⁡                                      (                                                                  2                        ⁢                        π                        ⁢                                                                                                  ⁢                        nk                                            M                                        )                                                  ⁢                                  sin                  (                                                            n                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                      B                      ⁢                                                                                                s                          ⁡                                                      (                                                                                          2                                ⁢                                π                                ⁢                                                                                                                                  ⁢                                k                                                            M                                                        )                                                                                                                                        M                                    )                                            ]                                                          (        2        )                                          B          n                =                  2          ⁢                      V            CC                    ⁢                                    ∑                              k                =                0                                            M                -                1                                      ⁢                                                            Sign                  ⁡                                      (                                          s                      ⁡                                              (                                                                              2                            ⁢                            π                            ⁢                                                                                                                  ⁢                            k                                                    M                                                )                                                              )                                                                    n                  ⁢                                                                          ⁢                  π                                            [                                                sin                  ⁡                                      (                                                                  2                        ⁢                        π                        ⁢                                                                                                  ⁢                        nk                                            M                                        )                                                  ⁢                                  sin                  (                                                            n                      ⁢                                                                                          ⁢                      π                      ⁢                                                                                          ⁢                      B                      ⁢                                                                                                s                          ⁡                                                      (                                                                                          2                                ⁢                                π                                ⁢                                                                                                                                  ⁢                                k                                                            M                                                        )                                                                                                                                        M                                    )                                            ]                                                          (        3        )            
FIG. 4 shows a schematic view of an N-bit digital PWM converter, and FIG. 5 shows a schematic view of an N-bit digital word representing quantized signal Q and resulted quantization error, where error=Q−S. As shown in FIG. 4, the N-bit PWM converter includes a numerical quantized unit 301 and a D-to-T (D→T) convertor 302 for converting a digital value to a time pulse width. The maximum amplitude of input ramp signal S is defined as U. The quantization can be expressed as:Q=U×Bin  (4)Bin=b12−1+b22−2+b32−3 . . . +bn2−n  (5)The minimum resolution of the quantized signal is:
                    Lsb        =                  U                      2            N                                              (        6        )            
Referring to FIG. 3 and FIG. 5, the relation of maximum time-slot length 2π/M corresponding to the peak value U of input signal S and the relation of minimum level resolution Lsb mapped to minimum length resolution LLSB can be depicted in FIG. 6.
FIG. 7 shows a schematic view of quantization noise error VQ. As shown in FIG. 7, VQ=L×Vcc, where L is the difference between minimum quantization length LLSB and minimum quantization resolution of Lsb.
Assume that the PWM output amplitude is unity, i.e., 1, and the N-bit word is only for expressing positive input value. The range of the error length is:
                              L          ∈                      [                                                            L                  Lsb                                2                            ,                              -                                                      L                    Lsb                                    2                                                      ]                          ,                              V            CC                    =          1                                    (        7        )            The rms value of the quantization noise signal, ⊃1], is given by
                              V                      Q            ⁡                          (              rms              )                                      =                                            [                                                1                  T                                ⁢                                                      ∫                                                                  -                        T                                            /                      2                                                              T                      /                      2                                                        ⁢                                                            V                      Q                      2                                        ⁢                                          ⅆ                      θ                                                                                  ]                        0.5                    =                                    [                                                1                  T                                ⁢                                                      ∫                                                                  -                        T                                            /                      2                                                              T                      /                      2                                                        ⁢                                                                                                              L                          Lsb                          2                                                ⁡                                                  (                                                                                    -                              θ                                                        T                                                    )                                                                    2                                        ⁢                                          ⅆ                      θ                                                                                  ]                        0.5                                              (        8        )                                                          ⁢                  =                                                    [                                                                            L                      Lsb                      2                                                              T                      3                                                        ⁢                                      (                                                                                            θ                          3                                                3                                            ⁢                                              ❘                                                                              -                            T                                                    /                          2                                                                          T                          /                          2                                                                                      )                                                  ]                            0.5                        =                                          L                LSB                                            12                                                                        (        9        )            Therefore, Quantization noise intensity rms is represented as:
                              V                      Q            ⁡                          (              rms              )                                      =                                            L              Lsb                                      12                                =                                    2              ⁢              π                                                      2                N                            ⁢              M              ⁢                              12                                                                        (        10        )            
FIG. 8 shows a schematic view of comparison between a differential PWM-DAC and a sample-and-hold DAC. As shown in FIG. 8, the differential PWM-DAC outputs digital pulse and the sample-and-hold DAC outputs analog signal. The SNR of the PWM-DAC is derived as:
                              S          ⁢                                          ⁢          N          ⁢                                          ⁢          R                =                              20            ⁢                          log              ⁡                              (                                                      Signal                    rms                                                        Noise                    rms                                                  )                                              =                                    20              ⁢                              log                (                                                                            1                      2                                        ⁢                                                                                            A                          1                          2                                                +                                                  B                          1                          2                                                                                                                                                2                      ⁢                      π                                                                                      2                        N                                            ⁢                      M                      ⁢                                              12                                                                                            )                                      =                          20              ⁢                              log                (                                                                            2                      N                                        ⁢                                          MC                      1                                        ⁢                                          12                                                                            4                    ⁢                    π                                                  )                                                                        (        11        )                                                      S            ⁢                                                  ⁢            N            ⁢                                                  ⁢            R                    =                                    6.02              ⁢              N                        +                          20              ⁢                              log                ⁡                                  (                                      MC                    1                                    )                                                      -                          11.18              ⁢                                                          ⁢              dB                                      ⁢                                  ⁢        Where                            (        12        )                                          C          1                =                                            A              1              2                        +                          B              1              2                                                          (        13        )                                          A          1                =                              ∑                          k              =              0                                      M              -              1                                ⁢                                                    2                ⁢                                  Sign                  ⁡                                      (                                          s                      ⁡                                              (                                                                              2                            ⁢                            π                            ⁢                                                                                                                  ⁢                            k                                                    M                                                )                                                              )                                                              π                        [                                          cos                ⁡                                  (                                                            2                      ⁢                      π                      ⁢                                                                                          ⁢                      k                                        M                                    )                                            ⁢                              sin                (                                                      π                    ⁢                                                                                  ⁢                    B                    ⁢                                                                                        s                        ⁡                                                  (                                                                                    2                              ⁢                              π                              ⁢                                                                                                                          ⁢                              k                                                        M                                                    )                                                                                                                            M                                )                                      ]                                              (        14        )                                          B          1                =                              ∑                          k              =              0                                      M              -              1                                ⁢                                                    2                ⁢                                  Sign                  ⁡                                      (                                          s                      ⁡                                              (                                                                              2                            ⁢                            π                            ⁢                                                                                                                  ⁢                            k                                                    M                                                )                                                              )                                                                              n                ⁢                                                                  ⁢                π                                      [                                          sin                ⁡                                  (                                                            2                      ⁢                      π                      ⁢                                                                                          ⁢                      k                                        M                                    )                                            ⁢                              sin                (                                                      π                    ⁢                                                                                  ⁢                    B                    ⁢                                                                                        s                        ⁡                                                  (                                                                                    2                              ⁢                              π                              ⁢                                                                                                                          ⁢                              k                                                        M                                                    )                                                                                                                            M                                )                                      ]                                              (        15        )            In contrast, the SNR for the sample-and-hold DAC is 6.02N+10 log(M)+1.76 dB.
The SNR of differential PWM-DAC is a function of quantization bit-N, over sample-rate M and input modulating signal band-width BW. FIG. 9 shows a schematic plot of the SNR corresponding input signal's BW and PWM sample-rate ωc=Mωs when N is set as 14 bits. As shown in FIG. 9, in order to maintain SNR>100 dB for differential PWM output with respect to audio band-width 20 Khz, where M=25, BW=20 Khz, N=14:
                              P          ⁢                                          ⁢          V          ⁢                                          ⁢          M          ⁢                                          ⁢          D          ⁢                                          ⁢          A          ⁢                                          ⁢          C          ⁢                                          ⁢          S          ⁢                                          ⁢          N          ⁢                                          ⁢          R                =                                            6.02              ⁢              N                        +                          20              ⁢                              log                ⁡                                  (                  M                  )                                                      -                          11.18              ⁢                                                          ⁢              dB                                =                                                    6.02                ×                14                            +                              20                ⁢                                  log                  ⁡                                      (                    25                    )                                                              -                              11.18                ⁢                                                                  ⁢                dB                                      =                          101              ⁢                                                          ⁢              dB                                                          (        16        )            
There is a critical choice for minimum-time-resolution (or minimum-time-slot) of differential PWM as shown in FIG. 7:
                              Minimum          ⁢                                          ⁢          Time          ⁢                                          ⁢          resolution          ⁢                                          ⁢                      T            R                          =                              1                                          2                N                            ⁢              M              ×              BW                                ⁢          sec                                    (        17        )                                          T          R                =                                            1                              8.192                ⁢                                                                  ⁢                Ghz                                      ⁢            sec                    =                      122            ⁢                                                  ⁢            ps                                              (        18        )            
As revealed in equation (16), the minimum-time-resolution should reach 122 ps to guarantee SNR greater than 100 dB. This is very short pulse-width for differential PWM implementation and may raise two issues. The first issue is how to generate this short pulse while lowering the power consumption and cost; and the second issue is that the next stage of differential PWM output is a power driver stage, which will cause the short pulse diminished when signal pass through the power driver due to the dead-time and power MOS's parasitic capacitor, as shown in FIG. 10.
Thus, it is imperative to devise a solution to address the aforementioned issues.